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XXXIV. — Detached Theorems on Circulants. By Thomas Muir, LL.D. 



(Read 4th May 1885.) 



1. If in a circulant the places (r, s) and (p, q), owing to the cyclical permuta- 

 tion, be occupied by the same element, then the complementary minor of this 

 element in its first place is to its complementary minor in the second as 

 (_l) r + 8 ;(_i)p+i. 



To prove that the complementaries are as ( — 1)' + ' :( — l) p+q is the same as 

 to prove that the cofactors are identical. And as the cofactor of an element of 

 a determinant is not altered by the transposition of rows and columns provided 

 the determinant itself is not thereby altered, it is evident that all we have to 

 show is that the element in the place (r, s) may by transposition of rows and 

 columns be made to take the place (p, q), and the determinant remain in out- 

 ward form the same as before. Now it is a known property of the circulant 

 that this can be done by first bringing the element in the place (r, s) into the 

 p th row by cyclical transposition of rows, and thereafter making an exact 

 similar set of transpositions of columns. Thus the theorem is established. 



2. If a,, b, c, . . . be the elements of the first row of a circulant, say of the fifth 

 order, and A, -B, C, — , . . . their complementaries, then the circulant is equal to 



(a + wb + w-c + w n d + w 4 eXA + co 4 B + co 3 C + W 2 D + toE) 



where o is any fifth root of unity. 



Multiplying the circulant 



or C (a b c d e), 



a 



b 



c 



d 



e 



e 



a 



b 



c 



d 



d 



e 



a 



b 



c 



c 



d 



e 



a 



b 



b 



c 



d 



c 



a 



by 1 in the form 



VOL. XXXII. PART III. 



d o 



